Show mesh vertices ID
There's a built-in feature to show mesh element indices, but the option isn't shown in the user interface by default.

Blender 2.7 and older:

The option only shows if the debug mode is enabled.


Start Blender in debug mode, or simply go to Python Console (Shift-F4) and type bpy.app.debug = True
Go to N-panel > Mesh Display
A checkbox Indices will now show
The colors are controlled by the interface theme


There's also an updated version of the Index Visulizer script:
space_view3d_index_visualiser_bmesh.py

Blender 2.8:

It only shows if the developer UI option is turned on.


Run Blender normally
Click Edit > Preferences...
In the Display panel of the Interface tab, tick Developer Extras
or type bpy.context.preferences.view.show_developer_ui = True into the Python console.
Open the Overlays popover in the 3D View 
Look for the label Developer and tick the checkbox Indices